Random particle wiggle on sphere
I think the solution is fairly simple and I will try to explain how I would do it. If needed I can add Images explaining everything as soon as I have access to a PC.
You need basically only one thing, you need the particles to be constraint to the surface of the sphere. There are two good ways of doing that.
At first you need to duplicate the emitter sphere and make the duplicate a collision object (physics tab) and set damping to 1.0, then you have two options for making the particles stay. The first option is to add a solidify modifier to the collision sphere and scale it so the particles spawn inside. I personally think the better more flexible way to do it is adding a forcefield to the duplicate collision sphere with negative strength to attract the particles.
For the random movement just add a turbulence force field or turn up the random initial velocity of the particles.

If you want your particles to stay fully on the sphere and not only half, turn on size deflect for the particles

Very important to know is that the object which is the particle emitter has no effect on the particles, no matter if you use collision, force field or something else.

If you want multiple spheres in your simulation, I would recommend using the solution with two collision spheres, because its easier to setup. It is still possible to setup the force fields in a way that the particles will stay on the surface, but It is a little bit more complicated. The advantage though is that one particle from one sphere can travel to touching other spheres, I personally think, that makes a good effect.